Alternatives to QFT

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To people familiar with QFT. You know quantum fields are non-interacting and they use perturbations methods. Is there other studies or programme that would replace conventional QFT with full fledged interacting quantum fields?

Also about Second Quantization where they treat the Klein-Gorden and Dirac equations acting like classical equations like Maxwell Equations and quantize them to create field quantas such as matter or fermionic fields. Is there any studies or programme about alternative to this? Or are you certain 100% that Second Quantization is fully correct?

And if QFT being not yet perfect due to the non-interacting fields for example. Why are physicists convinced they an arrive at the Theory Of Everything when the foundations are faulty... or maybe they are just contended for now to arrive at Quantum Gravity? And can one even reach it with a possibily faulty QFT foundations? Maybe there is no theory of quantum gravity precisely because QFT is faulty? How possible is this?
 

martinbn

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You impression of QFT is very inaccurate.
 
You impression of QFT is very inaccurate.
I learnt it from M.Y. Han's book "A Story Of Light: A Short Introduction To Quantum Field Theory Of Quarks And Leptons"

https://www.amazon.com/dp/9812560343/?tag=pfamazon01-20

Which part of the following do you think is inaccurate and why?

The first leap of faith is the introduction of the concept of matter fields, as discussed in Chapter 7. The quantization of the electromagentic field successfully incorporated photons as the quanta of that field and - this is critical - the electromagnetic field (the four-vector potential) satisfied a classical wave equation identical to the Klein-Gordon equation for zero-mass case. A classical wave equation of the 19th century turned out to be the same as the defining wave equation of relativistic quantum mechanics of the 20th century! This then led to the first leap of faith - the grandest emulation of radiation by matter - that all matter particles, electrons and positrons initially and now extended to all matter particles, quarks and leptons, should be considered as quanta of their own quantized fields, each to its own. The wavefunctions of the relativistic quantum mechanics morphed into classical fields. This conceptual transition from relativistic quantum mechanical wavefunctions to classical fields was the first necesary step toward quantized matter fields. Whether such emulation of radiation by matter is totally justifiable remains an open question. It will remain an open question until we successfully achive completely satisfactory quantum field theory of matter, a goal not yet fully achieved.
 
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So, where does he say matter fields are non-interacting?
 

tom.stoer

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there are of course fully non-perturbative methods in QFT
 
So, where does he say matter fields are non-interacting?
It's in another chapter but I learnt it first in the quantum physics forum by Fredrik who says Fock Space in QFT is non-interacting (which of the following is inaccurate, please correct it):

"A Fock space is constructed from the Hilbert space associated with the single-particle theory. You use the single-particle space to construct a space of 2-particle states, a space of 3-particle states, and so on, and then you combine them all into a Hilbert space that contains all the 1-particle states, all the 2-particle states, and so on. This Hilbert space is called a Fock space. So it's just an algebraic construction. You need nothing more than the Hilbert space from the single-particle theory to define it, and the single-particle theory can be defined using a Lagrangian with no products of more than two field components or derivatives of field components.

However, in non-rigorous QFT, I think the idea is just to ignore that the interacting Hilbert space is really a different Hilbert space, and just introduce operators that can take n-particle states to (n+1)-particle states for example. In this context, Fock space is, as you put it, "pretending to have interaction when it doesn't really". I really suck at QFT beyond the most basic stuff, so I can't explain it better, and I might even be wrong (about the stuff in this paragraph)."
 
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It's in another chapter but I learnt it first in the quantum physics forum by Fredrik who says Fock Space in QFT is non-interacting (which of the following is inaccurate, please correct it):

"A Fock space is constructed from the Hilbert space associated with the single-particle theory. You use the single-particle space to construct a space of 2-particle states, a space of 3-particle states, and so on, and then you combine them all into a Hilbert space that contains all the 1-particle states, all the 2-particle states, and so on. This Hilbert space is called a Fock space. So it's just an algebraic construction. You need nothing more than the Hilbert space from the single-particle theory to define it, and the single-particle theory can be defined using a Lagrangian with no products of more than two field components or derivatives of field components.

However, in non-rigorous QFT, I think the idea is just to ignore that the interacting Hilbert space is really a different Hilbert space, and just introduce operators that can take n-particle states to (n+1)-particle states for example. In this context, Fock space is, as you put it, "pretending to have interaction when it doesn't really". I really suck at QFT beyond the most basic stuff, so I can't explain it better, and I might even be wrong (about the stuff in this paragraph)."
I must admit this is the first time I hear of a Hilbert space being interacting or not. You may claim that the basis vectors constructed as a direct product of single-particle kets of arbitrary power are eigenkets of the Hamiltonian of the system only when the theory is non-interacting, but the space spanned by them is independent of the basis, and, at least in principle, one should be able to diagonalize even the interacting Hamiltonian acting on kets in this Fock space.

In my opinion, the most important sentence in your post is the bolded one. If there are products of more than 2 field operators in the Lagrangian, then this is necessarily an interction.
 
I must admit this is the first time I hear of a Hilbert space being interacting or not. You may claim that the basis vectors constructed as a direct product of single-particle kets of arbitrary power are eigenkets of the Hamiltonian of the system only when the theory is non-interacting, but the space spanned by them is independent of the basis, and, at least in principle, one should be able to diagonalize even the interacting Hamiltonian acting on kets in this Fock space.

In my opinion, the most important sentence in your post is the bolded one. If there are products of more than 2 field operators in the Lagrangian, then this is necessarily an interction.

What? Let's go to the context used by M.Y. Han book "A Story Of Light: A Short Introduction To Quantum Field Theory Of Quarks And Leptons".

I'll quote only the relevant passages and omit the math and other detailing part:

"The quantization of fields and the emergence of particles as quanta of the quantized fields discussed in Chapter 9 represent the very essence of quantum field theory. The fields mentioned so far - Klein-Gorden, electromagnetic as well as Dirac fields - are, however, only for the non-interacting cases, that is, for free fields devoid of any interactions, the forces. The theory of free fields by itself is devoid of any physical content: there is no such thing in the real world as a free, non-interacting electron that exerts no force on an adjacnet electron. The theory of free fields provides the foundations upon which one can build the framwork for introducing real physics, namely, the interaction among particles."

[omitting 2 pages of calculations and details]

"Quantum field theory for interacting particles would have been completely solved, and we could have moved beyond it. Well, not exactly. Not exactly, because no one can solve the highly nonlinear copuled equations for interacting fields that result from the interacting Lagrangian density obtained by the subtitution rule. Exact and analytical solutions for interacting fields have never been obtained: we ended up with the Lagrangian that we could not solve!"

[omitting a page]

"At this point, the quantum field theory of interacting particles proceeded towards the only other alternative left: when so justified, treat the interaction part of the Lagrangian as a small perturbatoin to the free part of the Lagrangian"

[I won't quote other paragraphs anymore. Just see it in amazon free page preview if necessary]

Do you know the part about "subtitution rule" he was talking about? Any relation to it that you are talking about? He basically said the subtitution rule couldn't be solved. And we are left only with perturbation, and we know it is seems ad hoc. Therefore Quantum Field Theory seems to be flawed. How then could they arrive at the right theory of Quantum Gravity with such a flawed foundation?!
 

atyy

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For a simple analogy, a linear oscillator has sinusoidal oscillations.

A nonlinear oscillator does not have sinusoidal oscillations.

Can the solution to the nonlinear oscillator be expressed as a sum of sinusoidal oscillations? Yes - that's what Fourier decomposition is.

For QFT, the analogy is:
linear -> non-interacting
nonlinear -> interacting
sinusoidal -> Fock space.
 

tom.stoer

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The Fock space states are "blind" for interactions. The interactions are represented by
operators acting on Fock states. It's true that for some questions Fock states are not the best calculational tool, but they are not a foundational problem.
 
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To people familiar with QFT. You know quantum fields are non-interacting and they use perturbations methods. Is there other studies or programme that would replace conventional QFT with full fledged interacting quantum fields?

Also about Second Quantization where they treat the Klein-Gorden and Dirac equations acting like classical equations like Maxwell Equations and quantize them to create field quantas such as matter or fermionic fields. Is there any studies or programme about alternative to this? Or are you certain 100% that Second Quantization is fully correct?

And if QFT being not yet perfect due to the non-interacting fields for example. Why are physicists convinced they an arrive at the Theory Of Everything when the foundations are faulty... or maybe they are just contended for now to arrive at Quantum Gravity? And can one even reach it with a possibily faulty QFT foundations? Maybe there is no theory of quantum gravity precisely because QFT is faulty? How possible is this?
i)
Only free fields are well-defined in QFT, but there is not a replacement for «fully fledged interacting quantum fields» because the concept of field is not defined there.

ii)
«Second Quantization» is a misnomer. There is nothing that is quantized twice as Weinberg often remarks. 'Second' quantization is a formalism for dealing with creating/destruction and creation/destruction is also used in ordinary QM.

iii)
Only some naive physicists as string (brane and M) theorists believed that they could obtain a «Theory Of Everything» over the basis of QFT.

Others are working in more general theories, including far reaching generalizations of string, brane, and M theory.

iv)
The fiasco with quantum gravity has little to see with the limitations of QFT, and more with misunderstandings about general relativity.
 

tom.stoer

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i)
Only free fields are well-defined in QFT, but there is not a replacement for «fully fledged interacting quantum fields» because the concept of field is not defined there.
Agreed -
- with a minor comment or question: is it really the concept of a "field" or a "field operator" that makes problems, or the concept for "interaction of fields".

ii)
«Second Quantization» is a misnomer. There is nothing that is quantized twice as Weinberg often remarks. 'Second' quantization is a formalism for dealing with creating/destruction and creation/destruction is also used in ordinary QM.
Agreed

iii)
Only some naive physicists as string (brane and M) theorists believed that they could obtain a «Theory Of Everything» over the basis of QFT.
Personally I agree that string- or M-theory may not the final answer, but I would not dare to call them naive, as long as I have nothing else to offer.

iv)
The fiasco with quantum gravity has little to see with the limitations of QFT, and more with misunderstandings about general relativity.
Here I don't agree; I think that in many approaches to quantum gravity one takes general relativity quite seriously (e.g. LQG); and I think that even reserach programs that are more inspired by QFT methods (string theory, asymptotic safety) do take GR seriously.
 
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Any relation to it that you are talking about? He basically said the subtitution rule couldn't be solved. And we are left only with perturbation, and we know it is seems ad hoc. Therefore Quantum Field Theory seems to be flawed. How then could they arrive at the right theory of Quantum Gravity with such a flawed foundation?!
I'm afraid you had performed a logical fallacy here. Namely, your conclusion, "QFT seems to be flawed", does not follow from the premises you gave. Namely, "we know it seems ad hoc" does not count as logical reasoning.

Then, your last question is a false contradiction
(
It is equivalent to the line of reasoning:
1. If we can arrive at Quantum Gravity with the current formalism, then we know the current formalism is correct.
2. QFT is part of the current formalism.
-----------------------------------------------------------------------------------------
If we know the current formalism is correct, then we know QFT is correct.
If we can arrive at QG with the current formalism, then QFT is correct.
3. QFT is incorrect.
-------------------------------------------------------------------------------------
We cannot arrive at QG with the current formalism.
)
because premise 3 is the wrong conclusion that you drew from the above wrong analysis.

Also, relating to my previous post, see tom.stoer's post #10:

The Fock space states are "blind" for interactions. The interactions are represented by
operators acting on Fock states. It's true that for some questions Fock states are not the best calculational tool, but they are not a foundational problem.
 
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Here I don't agree; I think that in many approaches to quantum gravity one takes general relativity quite seriously (e.g. LQG); and I think that even research programs that are more inspired by QFT methods (string theory, asymptotic safety) do take GR seriously.
They take GR seriously, but what I said is that they misunderstand GR.

String theorists were notorious for believing that GR is equivalent to a spin-2 field theory over a flat background. And claimed that string theory was the final theory. String theorists did need about 40 years to understand that they would begin to search a background-less version (M-theory), but no string theorist has serious ideas about what M-theory is (M is somewhat used for Mistery).

LQG community is also rather confused but in a somewhat complementary way.
 

tom.stoer

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String theorists were notorious for believing that GR is equivalent to a spin-2 field theory over a flat background. And claimed that string theory was the final theory. String theorists did need about 40 years to understand that they would begin to search a background-less version (M-theory), but no string theorist has serious ideas about what M-theory is (M is somewhat used for Mistery).
I can agree to that view.

LQG community is also rather confused but in a somewhat complementary way.
I commented on this confusion (from my persepctive) in some other threads.
 
I learnt from M.Y. Han book that there are 3 phases of development of quantum field theory and how they deal with non-interacting fields. I'll summarize it.

First phase (Early 1950s) - Langrangian Field Theory - based on canonical quantization, success in QED followed by non-expandability in the case of strong nuclear force and by non-renomalizability in the case of weak nuclear force.

Second phase (1950s-1960s) - Axiomatic QFT - for example S-Matrix theories and other axiomatic approaches, however they did not bring solutions to quantum field theories any closer than the Lagrangian field theories.

Third phase (1970s) - (Lagrangian) gauge field theory - ongoing

My question is. Can you make use of Gauge Theory without using Quantum Field Theory? Or the two completely related? But noether theorem can be applied to newtonian physics so can the gauge symmetry concept of electromagnetism U(1), electroweak U(1)xSU(2), Strong SU(3) can be developed without using the concept of quantum field theory?
 
2,956
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I learnt from M.Y. Han book that there are 3 phases of development of quantum field theory and how they deal with non-interacting fields. I'll summarize it.

First phase (Early 1950s) - Langrangian Field Theory - based on canonical quantization, success in QED followed by non-expandability in the case of strong nuclear force and by non-renomalizability in the case of weak nuclear force.

Second phase (1950s-1960s) - Axiomatic QFT - for example S-Matrix theories and other axiomatic approaches, however they did not bring solutions to quantum field theories any closer than the Lagrangian field theories.

Third phase (1970s) - (Lagrangian) gauge field theory - ongoing

My question is. Can you make use of Gauge Theory without using Quantum Field Theory? Or the two completely related? But noether theorem can be applied to newtonian physics so can the gauge symmetry concept of electromagnetism U(1), electroweak U(1)xSU(2), Strong SU(3) can be developed without using the concept of quantum field theory?
So, if I understood your exposition correctly, the third phase of the development of quantum field theory is gauge field theory. And, then, you ask if we can use gauge theory without using quantum field theory? Does this even make sense?
 

strangerep

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I must admit this is the first time I hear of a Hilbert space being interacting or not.
A more accurate phrase is "supports an interacting representation of the Poincare group".

[Waterfall] may claim that the basis vectors constructed as a direct product of single-particle kets of arbitrary power are eigenkets of the Hamiltonian of the system only when the theory is non-interacting, but the space spanned by them is independent of the basis, and, at least in principle, one should be able to diagonalize even the interacting Hamiltonian acting on kets in this Fock space.
No, actually. This is the content of Haag's theorem. The basis eigenstates of a free Fock space fail (in general) to span the interacting Fock space. (In this sense, they are indeed a foundational problem.) That's one of the reasons why infinite renormalizations are necessary: they kinda "push" you toward the correct space (in perturbative steps). It's also part of the reason why rigorous proof of convergence for 4D QFTs remains problematic.
 
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strangerep

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[...] we are left only with perturbation, and we know it is seems ad hoc.
Actually, perturbation techniques in QFT are essentially a version of the Poincare-Lindstedt method in classical dynamics. E.g., if one tries to solve the quartic anharmonic oscillator using naive perturbation theory in terms of harmonic oscillator solutions, one gets solutions that escape to infinity. But the quartic anharmonic oscillator potential is clearly confining, so this cannot be correct. However, if instead of perturbation as a sequence of solutions of the harmonic oscillator, we regard it as a sequence of similar theories -- in which the coupling constants like mass and stiffness are also expanded perturbatively, then we get a much better solution involving higher harmonics. (The classical dynamics text of Jose & Saletan explains this reasonably well.)

In perturbative QFT with renormalization, we do something similar: the mass and other "constants" are considered as series expansions which we adjust at each perturbative order to eliminate any unphysical nonsense. It is, of course, remarkable that this technique of perturbation approximation as a "sequence of similar theories" yields results agreeing with experiment to extraordinary accuracy.

Therefore Quantum Field Theory seems to be flawed.
"Flawed" is far too harsh a word. More accurate is that "convergence of the perturbation series in 4D QFT has not been rigorously established". Also note that there is such a thing as "asymptotic series" in which the first few terms are excellent approximations, but the approximations then get worse for higher orders.

Also remember that QFTs are among the most accurate theories in all of physics, especially QED, in terms of agreement between theory and experiment.
 
there are of course fully non-perturbative methods in QFT
Generally? I've seen large N methods, but thats just perturbation theory in 1/N. Even the lattice is sort of an expansion in the lattice constant.
 

atyy

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No, actually. This is the content of Haag's theorem. The basis eigenstates of a free Fock space fail (in general) to span the interacting Fock space. (In this sense, they are indeed a foundational problem.) That's one of the reasons why infinite renormalizations are necessary: they kinda "push" you toward the correct space (in perturbative steps). It's also part of the reason why rigorous proof of convergence for 4D QFTs remains problematic.
I've generally heard that it just says the interaction picture doesn't exist.

So why not just use the Heisenberg picture to define the theory?

Then use the interaction picture for approximation.

Is it any different from condensed matter, which is just Schroedinger's equation, so the problems are not foundational, just a matter of approximation? There the single-particle states are often hoped to span the solution space. There's no guarantee of that, but it's ok as long as they give a good approximation to the interacting ground state. And sometimes one just has to guess a trial variational wavefunction like BCS or Laughlin.

It is, of course, remarkable that this technique of perturbation approximation as a "sequence of similar theories" yields results agreeing with experiment to extraordinary accuracy.
In the case of QED, isn't this explained by the infra-red fixed point in the renormalization group flow? But maybe that's the same, I recently read that the KAM theorem is actually a case of RG flow - have no idea how that's the case.

"Flawed" is far too harsh a word. More accurate is that "convergence of the perturbation series in 4D QFT has not been rigorously established". Also note that there is such a thing as "asymptotic series" in which the first few terms are excellent approximations, but the approximations then get worse for higher orders.
Isn't it thought that it shouldn't converge for QED, because of the Landau pole, unless the theory is asymptotically safe? QCD on the other hand is supposed to be UV complete. Since it's still a Clay problem, it obviously hasn't been proved, but I've seen references to bits of constructive field theory in QCD, such as Osterwalder-Schrader reflection positivity, which is one condition to establish the Wightman axioms.
 
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strangerep

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I've generally heard that it just says the interaction picture doesn't exist.
Yes, that's another of the heuristic ways to describe the importance of the theorem. The rigorous statements of it are a bit different of course. (The Wiki page is rather poor, imho.)

So why not just use the Heisenberg picture to define the theory?
Dirac advocated something related to this. Have you read his Yeshiva QFT lecture notes?

P.A.M. Dirac, "Lectures on Quantum Field Theory",
Belfer Graduate School of Science, Yeshiva Univ., NY, 1966

I found them very thought-provoking. The Heisenberg picture is indeed superior to the Schrodinger picture in some aspects, but doesn't solve all the problems.

Then use the interaction picture for approximation.
The difficulty (iiuc!) is that one still needs time-dependent vacuum states and associated GNS representations (which are unitarily inequivalent in general for different values of t). Thus, one must work in something larger than Fock space.

Is it any different from condensed matter, which is just Schroedinger's equation, so the problems are not foundational, just a matter of approximation? There the single-particle states are often hoped to span the solution space. There's no guarantee of that, but it's ok as long as they give a good approximation to the interacting ground state. And sometimes one just has to guess a trial variational wavefunction like BCS or Laughlin.
In condensed matter theory, Bogoliubov transformations are often lurking about, mapping between unitarily inequivalent representations. If one can find sufficiently nice Bogoliubov transformations then one may hope to diagonalize the Hamiltonian by successive such transformations. For the QFT case one must find "sufficiently nice" time-dependent B-transforms (to solve the full dynamics -- not merely the scattering problem).

In the case of QED, isn't this explained by the infra-red fixed point in the renormalization group flow?
Hmm. I haven't thought about it in that context. But the IR problems in QED are known to be eliminated at all orders of perturbation by a dressing of the fermions by certain photon coherent states (a la Chung, Kibble, and others). A closer examination of that dressing reveals that it's a Bogoliubov-like transformation to a unitarily-inequivalent representation.

Isn't it thought that it shouldn't converge for QED [...]
:-) That's why I included the safe phrase "asymptotic series" -- series which look like they're converging nicely for a while but then go haywire.
 
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I learnt from M.Y. Han book that there are 3 phases of development of quantum field theory and how they deal with non-interacting fields. I'll summarize it.

First phase (Early 1950s) - Langrangian Field Theory - based on canonical quantization, success in QED [...]
Except when you want to study a system so complex :rolleyes: as two fully interacting electrons... then as Weinberg remarks in his QFT book, the QFT theory of bound states is not still in a satisfactory shape. And for three electrons the situations is still poor.
 

atyy

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In condensed matter theory, Bogoliubov transformations are often lurking about, mapping between unitarily inequivalent representations. If one can find sufficiently nice Bogoliubov transformations then one may hope to diagonalize the Hamiltonian by successive such transformations. For the QFT case one must find "sufficiently nice" time-dependent B-transforms (to solve the full dynamics -- not merely the scattering problem).
I'm trying to learn what Haag's theorem is, and googling brings up articles by Fraser, and Earman and Fraser. It looks as if Haag's theorem only needs Euclidean invariance, so it would seem to apply to non-relativistic QFT. Does Haag's theorem apply in the non-relativistic QFT used in condensed matter? If Haag's theorem doesn't apply, is it because Euclidean invariance is broken by the lattice?
 
I learnt it from M.Y. Han's book "A Story Of Light: A Short Introduction To Quantum Field Theory Of Quarks And Leptons"

https://www.amazon.com/dp/9812560343/?tag=pfamazon01-20

Which part of the following do you think is inaccurate and why?
[STRIKE]Waterfall, that is a pop-sci book. You may learn about science from such texts, but never
the science itself. That's totally beyond their scope.[/STRIKE]

EDIT: That's me talking out of my posterior. After taking a closer look to the above text, it
looks more like a graduate-level introductory work. My humblest apologies.

Still, the specific way you frame your questions and your replies seems to indicate your
understanding of the basics is both insufficient and plagued with misconceptions. What's
your physics background? Your public profile doesn't say. The reason I'm asking is, as
harsh as this may sound, you need a knowledge of physics (and maths) equivalent to a
bachelor's degree in physics before having a chance to actually being able to learn QFT (from
actual science textbooks, that is). Note that you don't need an actual degree - just the
equivalent level of knowledge.
 
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